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Theorem vtocld 2606
 Description: Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
vtocld.1 (𝜑𝐴𝑉)
vtocld.2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
vtocld.3 (𝜑𝜓)
Assertion
Ref Expression
vtocld (𝜑𝜒)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem vtocld
StepHypRef Expression
1 vtocld.1 . 2 (𝜑𝐴𝑉)
2 vtocld.2 . 2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
3 vtocld.3 . 2 (𝜑𝜓)
4 nfv 1421 . 2 𝑥𝜑
5 nfcvd 2179 . 2 (𝜑𝑥𝐴)
6 nfvd 1422 . 2 (𝜑 → Ⅎ𝑥𝜒)
71, 2, 3, 4, 5, 6vtocldf 2605 1 (𝜑𝜒)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559 This theorem is referenced by:  funfvima3  5392  frec2uzzd  9186  frec2uzuzd  9188
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